Reverse mathematics, well-quasi-orders, and Noetherian spaces

TL;DR

This paper explores the relationship between well-quasi-orders and Noetherian spaces, showing that certain topological properties related to these orders are equivalent to the logical system ACA_0 in reverse mathematics.

Contribution

It introduces a new framework for analyzing second-countable topological spaces and establishes the equivalence of specific theorems to ACA_0 over RCA_0.

Findings

Theorems about Noetherian properties of topologies induced by well-quasi-orders are equivalent to ACA_0.

Goubault-Larrecq's result that upper topologies of $P(Q)$ are Noetherian is analyzed in reverse mathematics.

A new framework for second-countable topological spaces is developed for RCA_0.

Abstract

A quasi-order $Q$ induces two natural quasi-orders on $P(Q)$, but if $Q$ is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq showed that moving from a well-quasi-order $Q$ to the quasi-orders on $P(Q)$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on $P(Q)$ are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form "if $Q$ is a well-quasi-order then a certain topology on (a subset of) $P(Q)$ is Noetherian" in the style of reverse mathematics, proving that these theorems are equivalent to ACA_0 over RCA_0. To state these theorems in RCA_0 we introduce a new framework for dealing with second-countable topological spaces.